Polygon Proofs

I have been playing with an old activity recently that was shown to me in my first year of teaching about three different ways to prove the interior angle of a polygon. In short, it involves starting with each of the three diagrams below.

Starting with each of the three ways of looking at regular polygons, students are invited to work towards a general expression for the interior angle in terms of the number of sides.

This is obviously a lovely example of how different approaches can lead to the same result. Same start, same finish, different journey. The issue I have had with this over the years has been getting students to focus on the fundamental difference between these diagrams. In particular the 2 different ways that split the polygon into triangles. So I wanted to think about approaching it differently. I created the following diagram. It is all the regular polygons from triangle to decagon with the same side length.

I copied the image up to A3 and cut the triangles out. I gave groups of students a set of the triangles and asked them to make me all the regular polygons from triangle to decagon! This was not easy, but it did highlight that difference I was talking about. The hexagon and nonagon were easy because they were split in to equal triangles but the others provoked a lot more thought. I was pleased with the result!

Once students were done I gave out a copy of the diagram below and asked them to work out all of the angles in the picture. Again, I was happy with the activity and reasoning that resulted.

As with most classroom ideas, I had to try this one out to see what would happen and then see if has any impact on students ability to manage the proofs. I'll chew on it for a while and see what we come up with but I the idea has definitely shown potential for the kind of classroom activity I find the most productive!